Coefficient Stability and Linear Cumulants for Colored Interlacing Triangles

In a depth-two colored interlacing triangle, each color occurs twice on top and once below, in top-bottom-top order. We prove the coefficient-polynomiality and linear-cumulant conjectures of Blitvic and Petrov for depth-two colored interlacing triangles with arbitrary bottom row. At every fixed coefficient index k, the normalized coefficient is a polynomial in the number n of colors for n at least 2k+1, has leading coefficient 5^k/k!, and becomes integral after multiplication by k!. For k at least one the range is sharp; at n=2k the discrepancy from the stable polynomial is exactly 3^(k-1). A refinement by bottom-row inversions has leading coefficient (1+4u)^k/k! and gives a binomial limiting inversion law at fixed energy, together with a spatial uniform order-statistic limit for unit defects. The proof bounds the number of nonreset positions and obtains a formal simple-pole factorization with a degree-controlled remainder. We also establish all four explicitly conjectured low-degree formulas and eventual strict log-concavity at every fixed coefficient index. For identity bottom row, the sharp stability range improves to n at least k+1. These results answer Conjectures 4.3, 4.4 and 4.9 in Colored Interlacing Triangles and Genocchi Medians (SIGMA 22 (2026), 081, DOI 10.3842/SIGMA.2026.081).